论文标题
滑动和破裂
Slipping flows and their breaking
论文作者
论文摘要
研究了沿着具有滑动边界条件的刚体的不可压缩流的破裂过程。这种滑动流是可压缩的，这是为平行于刚性边框平行的速度分量梯度形成奇异性的主要原因。在二维和三维无粘性prandtl方程的框架中，分析研究滑动流。在两种情况下都发现了梯度灾难的标准。对于2D Prandtl方程，破裂都是针对沿边界和涡度梯度的平行速度进行的。对于三维prandtl流，即破裂，即在有限的时间内形成折叠，发生在速度梯度张量的对称部分以及反对称部分 - 涡度。在二维Euler方程的框架中，研究了两个平行板之间流量梯度形成速度梯度的问题。结果表明，根据双重指数定律，最大速度梯度随时间增长而增长，涡度梯度同时增加。仔细的分析表明，此过程不过是折叠，最大速度梯度与其宽度之间存在幂律关系：$％\ max | u_x | \ propto \ ell^{ - 2/3} $。
The process of breaking of inviscid incompressible flows along a rigid body with slipping boundary conditions is studied. Such slipping flows are compressible, which is the main reason for the formation of a singularity for the gradient of the velocity component parallel to rigid border. Slipping flows are studied analytically in the framework of two- and three-dimensional inviscid Prandtl equations. Criteria for a gradient catastrophe are found in both cases. For 2D Prandtl equations breaking takes place both for the parallel velocity along the boundary and for the vorticity gradient. For three-dimensional Prandtl flows, breaking, i.e. the formation of a fold in a finite time, occurs for the symmetric part of the velocity gradient tensor, as well as for the antisymmetric part - vorticity. The problem of the formation of velocity gradients for flows between two parallel plates is studied numerically in the framework of two-dimensional Euler equations. It is shown that the maximum velocity gradient grows exponentially with time on a rigid boundary with a simultaneous increase in the vorticity gradient according to a double exponential law. Careful analysis shows that this process is nothing more than the folding, with a power-law relationship between the maximum velocity gradient and its width: $% \max|u_x|\propto \ell^{-2/3}$.
